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Given the functions \( f(x) = 2x + 1 \) and \( g(x) = x^2 + 2x + 1 \), we need to find \( (f \circ g)(-2) \).

### Step-by-Step Solution

1. **Calculate \( g(-2) \)**:
   \[ g(x) = x^2 + 2x + 1 \]
   Substitute \( x = -2 \):
   \[ g(-2) = (-2)^2 + 2(-2) + 1 \]
   \[ = 4 - 4 + 1 \]
   \[ = 1 \]

2. **Calculate \( f(g(-2)) = f(1) \)**:
   \[ f(x) = 2x + 1 \]
   Substitute \( x = 1 \):
   \[ f(1) = 2(1) + 1 \]
   \[ = 2 + 1 \]
   \[ = 3 \]

Thus, 
\[ (f \circ g)(-2) = 3 \]

### Summary:
\[ (f \circ g)(-2) = 3 \]

Do you want details or have any questions?

### Related Questions:
1. What is \( (g \circ f)(-2) \)?
2. Find the inverse of \( g(x) \).
3. Calculate \( (f \circ g)(0) \).
4. Determine \( (f \circ f)(2) \).
5. What is the range of \( g(x) \)?
6. Find the derivative of \( f(x) \).
7. Calculate \( g(-1) \).
8. Determine the critical points of \( f(x) \).

### Tip:
Always check the domain of the inner function before substituting it into the outer function in composite functions.

Learn how to calculate (f ∘ g)(-2) for the functions f(x) = 2x + 1 and g(x) = x^2 + 2x + 1. This problem involves composite functions, quadratic and linear functions, and their composition. Suitable for high school students.

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